Non-Reflexive Logics Logics that Derogate the Standard Theory of Identity




Non-Reflexive Logics
Logics that Derogate the Standard Theory of
Identity
Décio Krause
Department of Philosophy
Federal University of Santa Catarina|
and
Graduate Program in Logic and Metaphysics
Federal University of Rio de Janeiro
Dedicated to Andrea Loparic, forerunner in exploring non-classical logics.
Abstract
In this paper an outline of a class of heterodox logics is presented.
These systems, roughly speaking, question the standard notion of iden-
tity as given by classical logic and standard mathematics. Due to the
apparent fundamentality of this notion, it is necessary to provide a strong
motivation for the elaboration of such systems, and perhaps the main one
comes from quantum mechanics. This link is mentioned, and some refer-
ences are made, but no details are presented; the references provide more
detailed works. We just describe some non-reflexive logics and mathemat-
ics, ending with a general discussion about the necessity of identity.
MSC2020: 03A10, 03B60, 81P10, 03G12.
PACS: 02.10.-v, 03.65.-w, 01.70.+w
Contents
1
Introduction
2
1.1
Identity
3
2
Characteristics of the standard theory of identity
5
3
Challenging identity
7
4
Non-reflexive logics
8
4.1
Quasi-set theory
10
Partially supported by CNPq.
1
5
Other non-reflexive logics
13
6
Is identity really fundamental?
13
7
Philosophy of non-reflexive logics
14
1
Introduction
The XXth century was the period of the raising of non-classical logics. Since
the most celebrated principles of classical logic are the principles of identity,
non-contradiction and the excluded middle, it has been mainly against these
rules that hererodox logics start being developed;1 really, it was most against
the two last ones. So, a roughly classification was advanced: paraconsistent
logics would be those logics in which the principle of non-contradiction is not
universally valid, and paracomplete logics are those systems where the principle
of the excluded middle is derogated. Today there are a plenty of paraconsistent
systems, so as of paracomplete logics of several kinds (intuitionistic logic and
many-valued logics being the main ones).
Although these three mentioned principles are not the only important ones
in classical logic, they become famous by historical reasons. Classical logic is
full of other ‘fundamental principles’, such as double negation, the explosion
rule and Peirce’s law. But our task here concerns the principle of identity.
Surprising enough, it is perhaps the less questioned principle of classical logic.
Today, we can say that we have free ourselves from the standard principles of
classical logic, with the possible exception of the principle of identity; this has
been, even today, a classical taboo. Why this is so? Some hints are advanced
bellow.
But, first of all, we need to agree that everyone of the mentioned principles
admit different and non equivalent formulations, and this is not different with
the principle of identity (from now on, simply PI).2
For instance, in the standard language of propositional calculus, we may put
p → p, or p ↔ p
(1)
being p a propositional variable. If we permit quantification over such vari-
ables, then ∀p(p → p) would be the case. In first-order languages, being x and
individual variable and F a unary predicate, we can write
∀x(F (x) → F (x)).
(2)
1There are also the (said) non-classical logics that complement classical logic with addi-
tional operators which provide its language a stronger capacity of expression, such as the
standard modal logics, deontic logics, temporal logics, and so on. These systems do not
question the classical principles, contrarily to the heterodox systems. Anyway, there are also
mixed systems, which are both complementary and heterodox, such as the paraconsistent
modal logics.
2The reader can find other formulations in [8].
2
Higher-order logics also encompass a version of PI, for instance,
∀F ∀x(F (x) → F (x)),
(3)
and so we go. But PI is not all that exists about identity. The most relevant
thing is what we can call the standard theory of identity (STI), which involves
more than the PI. Let us sketch STI a little bit bellow. Non-reflexive logics
are defined as those heterodox logics which violate STI, and in particular the
PI. But, first, it would be adequate to have a glimpse on the involved notion,
namely, identity.
1.1
Identity
‘Identity’ is an apparently simple notion. Googling it, we find things like “it
is the relation each thing bears only to itself”.3
Several other ‘definitions’ can
be find easily, so as the discussion whether identity would be a relation or
just a property. The notion looks simple, but hides insurmountable difficulties.
The discussions about this notion go back to the antiquity, mainly concerning
personal identity and identity through time (that is, the question of how can
we say that some thing that changes its properties can be said to be the same
thing as before). Some philosophers prefer to consider a different concept, yet
they continue to call it ‘identity’. This is the case of Peter Geach, who said that
only relative identity could have a sense; according to him, two things can be
said to be identical only relative to a certain ‘sortal’ predicate:4
x is the same
F as y, where F is a sortal predicate. But our account here is concerning logical
identity, or simply identity from now on.
In his Begriffsschrift [10], Frege assumed that the symbol of identity (dealt
with by the symbol of equality ‘=’, introduced by Robert Record in 1557) should
be put between two names, but later, he acknowledged in his On sense and
reference [11] that it must hold between objects. This is our understanding
up to now, although we can extend it to cope with properties, relations, and
functions as well. When we say that x = y is the case, or that it is true, we
mean that x and y denote the same thing, or that they have the same referent.
In other words, being this the case, there are no two things, but just one, which
can be named either as x or y (among possibly other means).
In nowadays logic, we have at our disposal different kinds of languages,
something never dreamed in Frege’s time: first-order languages, higher-order
languages, and the languages of set theories (I will leave the language of cat-
egories out of this discussion). In first-order languages, we have basically two
options to introduce STI: either we define identity or we take it as a primitive
notion. In order to define identity, we need a suitable formula of the language,
α(x, y), where x and y are free, and put
x = y := α(x,y)
(4)
4Sortal predicates, also called ‘count-nouns’ distinguish from ‘natural nouns’. So, if a sortal
predicate applies to some things, they can be named, ordered, counted.
3
For instance, we have Quine’s strategy of exhausting the (finitely many)
primitive predicates of the language [28]. In ZFC, axiomatized as a first-order
theory, we usually put
x = y := ∀z(x ∈ z ↔ y ∈ z),
(5)
and if there are atoms (ur-elements) (ZFU), we write
x = y := ∀z(x ∈ z ↔ y ∈ z) ∧ ∀z(z ∈ x ↔ z ∈ y).
(6)
Another alternative is to take ‘=’ as a primitive binary predicate subjected
to the following postulates:
1.
(Reflexivity) ∀x(x = x)
2.
(Substitutivity) ∀x∀y(x = y → (α(x) → α(y)), being y a term not encom-
passing x free.
In a first-order ZFC theory, we still add to these postulates the Axiom of
Extensionality in one of the forms bellow, depending whether there are atoms
or there are not:
∀x∀y(∀z(x ∈ z ↔ y ∈ z) ∧ ∀z(z ∈ x ↔ z ∈ y) → x = y)
(7)
or (without atoms)
∀x∀y(∀z(z ∈ x ↔ z ∈ y) → x = y)
(8)
From these postulates, it follows the symmetry and the transitivity of equal-
ity. In extensional contexts, such as ZFC, a binary relation is a set of ordered
pairs, and if identity is a binary relation, we can define the identity of the set
D as follows:
Definition 1.1 (The diagonal of a set) Let D be a non-empty set. The di-
agonal of D, or the identity of D, termed ΔD , is so defined:
ΔD := {x,x : x ∈ D}.
(9)
In semantic terms, let us recall, it is precisely in this set that the primitive
symbol ‘=’ is interpreted, being D the domain of the interpretation (but see
below).
Theorem 1.1 Let ∼ be an equivalence relation over the set D (in particular,
∼ can be a congruence). Then ΔD ⊆ ∼.
4
The proof is trivial, since every equivalence relation is reflexive. It is in this
sense that we say that the identity of D is the thiner equivalence relation (or
congruence) over a set.
In higher-order languages, we can define identity, usually by Leibniz’s Law.
Let us do it in second-order logic: being F a variable for predicates of individuals
and x and y individual variables, we have
Definition 1.2 (Identity in higher-order languages)
x = y := ∀F(F(x) ↔ F(y)).
(LL)
Informally speaking, the definition says that are equal (the same) those
entities that are indistinguishable relative to all their properties. Notice that
this theory entails that nothing more than properties (and relations) can be used
to characterize an identity. That is, we are in the field of the bundle theories
concerning identity and individuality (see [12] for a distinction between bundle
theories and substratum theories).
Interesting to say that LL is (of course in another way of expressing) the
definition given by Whitehead and Russell in their Principia Mathematica, and
was questioned by F. P. Ramsey, so as by L. Wittgenstein in the Tractatus. The
first said that there is no logical contradiction to suppose that there may be two
things which, despite indiscernible, are not the same thing [29, p.31]; the later
was more radical, thinking that the very notion of identity given by LL should
be ruled out since we can use one name for each object we are referring to (see
[24] and the references therein).
2
Characteristics of the standard theory of iden-
tity
The notion of identity characterized above by diverse means (as we have seen)
can be called ‘classical’. It entails several important facts, such as the following
ones:
1) The first-order classical identity doesn’t characterize the diagonal of the
domain up to elementary equivalence. Let us explain. By the first-order theory
of identity, we mean Reflexivity and Substitutivity, with ‘=’ as primitive. Now
let A = D, Ri, (i ∈ I), be a structure, with D = ∅ and the Ri being n-ary
relations over D, an interpretation to our language. This is what we call order-1
structure (see [23]), typical of Model Theory. Following W. Hodges, we call it a
structure with standard identity if ‘=’ is interpreted in the diagonal ΔD . But,
as Hodges shows, (a similar proof is given by Mendelson [27], who calls normal
these structures), there are structures which are elementary equivalent to A
that also model the postulates of identity. This means that from the point of
view of the language, we never know of what structures we are speaking about.
Consequently, the language doesn’t teach us whether we are speaking, say, of
objects of D or of subsets (equivalence classes) of elements of D.
5
This is of course not good for a theory which intends to be a theory of
something, say, the elements of D. In the informal parlance, we say that (first-
order) identity cannot be axiomatized, meaning that ΔD cannot be uniquely
captured from the first-order postulates of STI. The same, of course, holds
when the first-order identity is defined.
2) Thus, let us go to higher (second) order languages. Here, LL is mandatory.
Once we rule substratum out, we keep with properties and relations (and with
formulas in general), and then the collection of properties seems to be enough
to individuate an element of the domain, or to provide it its identity (although
individuality and identity are distinct concepts [25]). The semantics for our
second order language necessitates of a non-empty domain, as in the first-order
case, but also a collection of subsets of D where the unary predicate variables of
the language range, subsets of D ×D where the binary predicate variables range,
and do on. This is called a frame for our language. Notice that to individuate a
particular element of D, it is enough to have in the frame all unitary sets of the
elements of D. This kind of frame if called principal by Church [3, pp. 307-8].
The secondary interpretations take not all subsets of D, D × D, etc., but just
some of them. The consequences are huge.
First of all, with a principal interpretation, we lose completeness; with
secondary interpretation, we may have a weaker form of completeness, called
Henkin completeness. But notice that with a Henkin semantics, if not all sub-
sets of the domain are chosen, we may have a situation where x and y belong
to all the chosen subsets of the frame (that is, they obey all the same properties
of the language) and even so they are not necessarily the same object. That is,
Leibniz Law can fail. A simple example is enough. Suppose our second order
language has two individual constants a and b and three unary predicates P1,
P2, and P3. Let us take a frame with D = {1,2,3,4,5}, let a be interpreted in
3, b in 5 and P1, P2, P3 respectively in {1, 2, 3, , 5}, {3, 4, 5} and {1, 3, 5}. Then
Pi(a) ↔ Pj(b) for i,j = 1,2,3, but even so 3 = 5.
3) Let us suppose, finally, that we are with the STI in ZFC. Let a be any
set or atom. By the postulates, we can form the unitary set {a} and define
the following property we call the identity of a, namely, Ia(x) := x ∈ {a}.
Then a is the only object of the universe that has this property and therefore it
provides an identity criterion for a. If we call individual every object that has
an identity criterion, it results that every set or atom is an individual. Once
we can build all standard mathematics within ZFC, we can surely say that in
standard mathematics there are no solo numero indiscernible things, that is,
things that differ just for being two things, but without any further differences.
Just a remark: if the reader has a non trivial knowledge of set theory, prob-
ably she has heard about indiscernibles. There are different kinds of them (see
[33], [17] for instance). Apparently, guided by the word ‘indistinguishable’, these
entities are thought of as partaking all their properties (being indiscernible)
without being identical. But this is not strictly true. By the same argumenta-
tion given above, they are individuals since obey the postulates of ZFC, hence
of STI.
The way of dealing with indiscernibles within a standard framework such
6
as the ZFC set theory (the problem regarding category theory is still an open
problem) if to confine them to a deformable (non-rigid) structure, that is, a
structure encompassing other automorphisms than the identity function (the
trivial automorphism), as the ‘indiscernible’ of the previous paragraph. This
is a fake notion of an entity devoid of identity criterion, for it can be proven
[7] that every structure can be extended to a rigid structure (with just the
trivial automorphism) and so, if not in the original structure, we can discern
the elements of the domain in the extended structure. Since the whole universe
of sets (seen as a structure) is also rigid [17, p.66], we arrive to our result:
standard mathematics (and logic) is a theory of individuals, and in particular
some form of the principle of identity holds.
3
Challenging identity
In what sense STI can be questioned? The first way is to follow Ramsey: sim-
ply construct a logic where STI doesn’t hold. After all, as Hilbert has once
suggested, the mathematician should pursue all logically possible theories, and
not just those which approach reality (apud [20]). But this would be not neces-
sarily interesting. The better way would be to have some motivation for such a
logic. And here enters the more interesting point: there are plenty of reasons to
challenge identity in some way. Here we shall explore few, but make reference
to many others.
Alfred Korzybski was a Polish-American guy (a real Count) who in a certain
period has attended Tarski’s seminars in Berkeley. He became famous (appar-
ently, not for Tarski)5 for having written a book called Science and Sanity [19]
where, among other things, he claimed that nothing remains identical to itself
since it is always changing its properties, and that the lack of being aware of this
would be one of the problems with sanity of many people. Consequently, people
should to take into account non-Aristotelian habits, among then the rejection
of the Aristotelian sentence
A is A,
(10)
expressing his principle of identity, which should not hold in the real world.
Later, Oliver L. Reiser (1935) discussed what could be called non-Aristotelian
logics [30]. According to him, these would be logics violating one of the three
basic principles mentioned in the beginnings. Against the principle of non-
contradiction, he mentioned Hegelian metaphysics and the ‘dynamic’ logic of
Dewey. As for the excluded middle, he made references to Brouwer’s intuition-
ism and to the many-valued logics of Łukasiewicz, so as to Tarski and C. I.
Lewis. But, relatively to identity, Reiser mentions Korzybski. Unfortunately,
no logical system was stated, which happened only in 1948.
In this year, in the directions of the lines pointed out by Wittgenstein, O.
L. Zich (considered the father of Czech’s logic) presented a system without the
5In his book Fads and Fallacies in the Name of Science [14, Chap.23], Martin Gardner
doesn’t say much favoring Korzybski and his strange ‘semantics’.
7
symbol of identity but (as Wittgenstein suggested), writting a symbol for each
object in the domain of a supposed interpretation. According to Zich, LL of
Principia Mathematica could not be accepted, since it would be impossible to
check in a finite number of steps whether all the properties of two given objects
are the same. Zich’s paper was published in Polish, but there is a review by Jan
Kalicki [18].
I think that these are really the forerunners of the ideas of logics question-
ing the standard intuitive notion of identity as grounded in our metaphysical
tradition. As said before, the metaphysical notion that identity is something
an object share just to itself and with nothing more was impregnated in our
metaphysical pantheon. It would be difficult to leave this realm.
But, as we shall see below, perhaps the great inspiration for a logic without
identity (at least for some entities) comes from a quite reasonable interpretation
of the quantum mechanics formalism.
4
Non-reflexive logics
In his book Ensaio sobre os Fundamentos da Lógica [4], Newton da Costa in-
tended to show that every principle of classical logic can be questioned or,
as he prefers to say following Bachelard, dialetized. As for the principle of
identity, without any reference to these previous mentioned works, he found a
quite strong motivation in some philosophical ideas of the great physicist Erwin
Schrödinger, one of the father founders of quantum mechanics and Nobel win-
ner of 1933. In his book Science and Humanism [31], Schrödinger affirmed that
the notion of identity (‘sameness’) doesn’t apply to elementary particles in the
quantum realm. According to him, there is no sense in saying that a particle
here and now is the same as the particle there and after, despite everything
may suggest the opposite. Thus he runs in the same direction as David Hume,
who attributed to the re-identification of a thing as something due to the habit,
since according to him there is no causal connection between the first and the
second appearances of the object [16, passim].
Da Costa showed how to develop a first-order two sorted system he called
‘Schrödinger Logic’, denoted by ‘S’, as follows. The language has individual
variables and individual constants of two kinds. These are the terms of the
language. To the terms of the first kind, expressions of the kind s = t are not
formulas (the same for their negations). Thus, identity doesn’t apply to the
objects denoted by the terms of the first species. The axioms are immediate,
respecting that the theory of identity (really, STI), holds only for the objects of
the second kind (see [4, pp.117ff]).
We can repute S as the real first logical system where the principle of identity
in the first-order language (namely, ∀x(x = x)) is violated. Notice that da Costa
didn’t show that the negation of this sentence holds, for this would imply the
existence of an object that is not identical with itself (identity understood in
the standard sense). What he did was to suspend the judgment about that: the
notion of identity, as suggested by Schrödinger, simply fails to apply to some
8
objects: x = y is not always a well formed formula.
As for semantics, da Costa has just indicated a possible direction. He as-
sumed a set D as the domain, such that D = D1 ∪ D2 and interpreted the terms
of the first kind in D1 and the terms of the second kind in D2. But, as he
advanced, there are difficulties:
…D1 cannot be considered as a set in the standard sense of set
theories, since for its elements the relation of identity should have
no sense; only for the elements of D2 we can say that they are equal
[identical] or distinct. (op.cit., p. 119)
Then, and fundamentally, he suggests that
[i]n order to surpass this difficulty, there are two open ways: 1. to
look for a generalization of the notion of set, for instance build-
ing a theory of quasi-sets which would contain the standard sets as
particular cases, and in such a theory to edify a semantics for S.
[the second point concerns an informal semantics and shall be not
considered here].
This is really amazing. Not only a system of non-classical logic was sketched,
its semantics indicated, but a strong connection with quantum physics was in-
dicated. To explore these connections was the Ph.D. program developed by this
author under the supervision of da Costa in the late 1980s at USP (University of
São Paulo). In 1990, my Ph.D. thesis was approved [20]. There, I extended da
Costa system S to a higher-order logic Sω (simple theory of types) and provided
it a Henkin semantics according to which the system resulted Henkin-complete
(see also [5]) — this semantics was built in a standard set theory encompassing
STI. Furthermore, I went further in the ideas of Schrödinger and of quantum
mechanics in general, and developed a first version of the suggested theory of
quasi-sets [21].
But the remarks made above about the set D1 remained in my mind. It
would be necessary fo find a quasi-set semantics for Sω , so making semantics
in agreement with the claims of the logic. This was achieved in 1995 in my
thesis for full professor of Foundations of Mathematics at the Department of
Mathematics of the Federal University of Paraná, where I worked that time.
But things changed also with the logic. The used system was not the original
Sω, but a different one, a modal higher-order logic in the sense of Daniel Gallin
[13]. The reasons for choosing such a modal version was to consider, as M. L.
Dalla Chiara and G. Toraldo di Francia, that the quantum world seems to be a
“world of intensions” (see [9], [22], [6]), and then a purely extensional semantics
would not be well suited for this case.
In the quasi-set semantics, we may have (say) unary predicate constants
which are semantically associated not to a specific sets as in the extensional
semantics, but to some quasi-set (see below) of a collection of indiscernible
quasi-sets. So, we could give a formal account to the idea (expressed also by
Dalla Chiara and Toraldo di Francia, but see the above references) that, contrary
9
to standard semantics, in quantum mechanics one and the same intension may
have different ‘extensions’; as they exemplify, the ‘intension’ that characterize
electrons (a certain mass, electric charge, magnetic momentum, etc.) may have
different ‘extensions’, namely, any collection of electrons.
The system is a modal higher-order logic with a semantics given in the theory
of quasi-sets. Thus, we could vindicate the idea that some individual constants
can be associated to one element of a collection of indistinguishable objects, but
without the means to specify the very identity of this element (by the way, it
this kind of thing that happens in quantum mechanics when we say that one of
the two electrons of a Helium atom in the fundamental state has spin UP in a
given direction, while the another one has spin DOWN — nothing can tell us
which electron is this one, although it is described perfectly well by a definite
description :
‘the electron that has spin up in the chosen direction’). In the
same vein, we can grant that (say) a unary predicate might be associated to
‘everyone’ among a certain collection of indiscernible quasi-sets, precisely in the
direction advanced by Dalla Chiara and Toraldo di Francia. The system results
also Henkin-complete relative to such a semantics.
By the way, it seems to me that this was the first time that a semantics was
provided for a logical system using a non-standard theory of ‘sets’. This system
was also presented in [6].
4.1
Quasi-set theory
As we see, the theory of quasi-sets plays an important role in all of this. The
theory is a set-theoretical version of a non-reflexive logic, and roughly speaking
runs as follows, and surely is the strongest non-reflexive system we have till now.
The main target, as it were, is to deal with collections of indiscernible elements,
but without any standard trick of confining them to deformable (non-rigid)
structures or something similar. As suggested by the philosopher of physics
Heinz Post, at least in the quantum realm, indiscernibility should be looked for
“right from the start” (see [12] for all the discussion).
Let us call Q the theory of quasi-sets. Indiscernibility is a primitive concept,
dealt with by a binary relation ‘≡’ satisfying the properties of an equivalence
relation, but not full substitutivity.6 In this notation, ‘x ≡ y’ means ‘x is indis-
cernible from y’. This binary relation is a partial congruence in the following
sense: for most relations, if R(x, y) and x ≡ x, then R(x, y) as well (the same
holds for the second variable). The only relation to which this result does not
hold is membership: x ∈ y and x ≡ x does not entail that x ∈ y (for the poof,
see [12]).
Quasi-sets can have as elements other quasi-sets; particular quasi-sets (qsets),
termed sets, are copies of the sets in a standard theory (in the case, the Zermelo-
Fraenkel set theory with the Axiom of Choice).The theory admits also two kinds
of atoms (entities which are not sets), termed M -atoms (representing objects of
6If we add substitutivity to the postulates, then no differences between indiscernibility and
logical first-order identity would be achieved.
10
classical physics), which are copies of a standard set theory with atoms (ZFA)
and m-atoms (for quantum objects), which have quantum objects as their in-
tended interpretation, to whom it is supposed that the logical notion identity
(STI) does not hold. If we eliminate the m-atoms, we are left with a copy of
ZFU, the Zermelo-Fraenkel set theory with atoms. Hence, we can reconstruct
all standard mathematics within Q in such a ‘classical part’ of the theory.
Functions cannot be defined in the standard way. When m-atoms are present,
a standard function would not be able to distinguish between indiscernible ar-
guments and values. Therefore, the theory generalizes the concept to ‘quasi-
functions’ (‘q-functions’), which map indiscernible elements into indiscernible
elements.
Cardinals (termed ‘quasi-cardinals’) are also taken as primitive, although
they can be proven to exist for finite qsets (finite in the usual sense). The
concept of quasi-cardinals can be used to speak of ‘several objects’, even without
identity. So, when we say that we have two indiscernible q-functions, according
to the above definition, we are saying that we have a qset whose elements are
indiscernible q-functions and whose q-cardinal is two.7
The same happens in
other situations.
An interesting fact is that qsets composed of several indistinguishable m-
atoms do not have an associated ordinal. This lack of an ordinal means that
these elements cannot be counted by standard means, since they cannot be
ordered. However, we can still speak of a collection’s cardinal, its quasi-cardinal.
This existence of a cardinal but not of an ordinal is similar to what we have
in quantum physics when we say that we have some quantity of systems of the
same kind but cannot individuate or count them, e.g., the six electrons in the
level 2p of a Sodium atom.8
Identity (termed extensional identity) “=E ” is defined for qsets having the
same elements (in the sense that if an element belongs to one of them, then it
belongs to the another) or for M -objects belonging to the same qsets.9 However,
one can hypothesize that if a specific object belongs to a qset, then so and so.
This is similar to Russell’s use of the axioms of infinite (I) and choice (C) in
his theory of types, which assume the existence of certain classes that cannot
be constructed, so going against Russell’s constructibility thesis.
What was Russell’s answer? He transformed all sentences α whose proofs
depend on these axioms into conditionals of the form I
→ α and C → α.
Hence, if the axioms hold, then we can get α. In Q we are applying a similar
reasoning: if the objects of a qset belong to the other and vice-versa, then they
are extensionally identical. It should be noted that the definition of extensional
7Quasi-cardinals turn to be sets, so we can use the equality symbol among them. We use
the notation qc(x) = n for ‘the quasi-cardinal of (the qset) x’.
8To count a finite number of elements, say 4, is to define a bijection from the set with these
elements to the ordinal 4 = {0, 1, 2, 3}. This counting requires that we identify the elements
of the first set.
9There are subtleties that require us to provide further explanations. In Q, you cannot do
the maths and decide either a certain m-object belongs or not to a qset; this requires identity,
as you need to identify the object you are referring to. So, the theory employs some artifices
to deal with these situations.
11
identity holds only for sets and M -objects. It can be proven that the extensional
identity has all the properties of classical logical identity for the objects to which
it applies. However, it does not make sense for q-objects. That is, x =E y does
not have any meaning in the theory if x or y are m-objects. It is similar to
speak of categories in the Zermelo-Fraenkel set theory (supposed consistent).
The theory cannot capture the concept, yet it can be expressed in its language.
From now on, we shall abbreviate “=E ” by “=,” as usual.
The postulates of Q are similar to those of ZFA, but by considering that
now we may have m-objects. The notion of indistinguishability is extended to
qsets through an axiom that says that two qsets with the same q-cardinal and
having the same ‘quantity’ (we use q-cardinals to express that) of elements of
the same kind (indistinguishable among them) are also indiscernible. As an
example, consider two sulfuric acid molecules H2SO4. They can be seen as two
indistinguishable qsets, for both contain q-cardinal equals to 7 (counting the
atoms as basic elements), and the elements of the sub-collections of elements of
the same kind are also of the same q-cardinal (2, 1, and 4 respectively). Then we
can state that “H2SO4 ≡ H2SO4,” but of course, we cannot say that “H2SO4 =
H2SO4,” as for in the latter, the two molecules would not be two at all, but just
the same molecule (supposing, of course, that “=” stands for classical logical
identity). In the first case, notwithstanding, they count them as two, yet we
cannot say which is which (and this is the core idea).
Let us speak a little bit more about quasi-functions. Since physicists and
mathematicians may want to talk about random variables over qsets as a way to
model physical processes, it is important to define functions between qsets. This
can be done straightforwardly, and here we consider binary relations and unary
functions only. Such definitions can easily be extended to more complicated
multi-valued functions. A (binary) q-relation between the qsets A and B is
a qset of pairs of elements (sub-collections with q-cardinal equals 2), one in
A, the other in B.10
Quasi-functions (q-functions) from A to B are binary
relations between A and B such that if the pairs (qsets) with a and b and
with a and b belong to it and if a ≡ a, then b ≡ b (with a’s belonging
to A and the b’s to B). In other words, a q-function maps indistinguishable
elements into indistinguishable elements. When there are no m-objects involved,
the indistinguishability relation collapses in the extensional identity, and the
definition turns to be equivalent to the classical one. In particular, a q-function
from a “classical” set such as {1, −1} to a qset of indiscernible q-objects with
q-cardinal 2 can be defined so that we cannot know which q-object is associated
with each number (this example will be used below).
To summarize, in this section, we showed that the concept of indistinguisha-
bility, which conflicts with Leibniz’s Principle of the Identity of Indiscernibles,
can be incorporated as a metaphysical principle in a modified set theory with in-
distinguishable elements. This theory contains “copies” of the Zermelo-Fraenkel
axioms with Urelemente as a particular case when no indistinguishable q-objects
10We are avoiding the long and boring definitions, as, for instance, the definition of ordered
pairs, which presuppose lots of preliminary concepts, just to focus on the basic ideas. For
details, the interested reader can see the indicated references.
12
are involved. This theory will provide us the mathematical basis for formally
talking about indistinguishable properties, which we will show can be used in a
theory of quantum properties. We will see in the next section how we can use
those indistinguishable properties to avoid contradictions in quantum contextual
settings such as KS.
5
Other non-reflexive logics
To derogate the quantificational principle of identity is not the only way to
build non-reflexive logics. It can be done also at the propositional level, as the
following case illustrates.
In the propositional level, as we have seen, the principle of identity can be
written as p → p, where p is a propositional variable, and this is one of the thesis
of classical propositional logic. But, as shown by R. Sylvan and da Costa, this
can also be challenged [32]. In their logic, termed logic of causal implication,
a kind of conditional ∋ is used, so that A ∋ B means A causes B, being A
and B not sentences, but certain terms which “are sometimes propositional or
fact-like or rendered such by happening or occurrence functions” (op.cit.). The
authors claim that they do not intend to explain causation, a difficult topic
which perhaps cannot be covered by a sentential connective, but just to deal
with the idea that something may implies something, as (their example) ‘that
flood caused a famine’, and leave open the possibility of the introduction of
other ideas such as ‘reverse causation’, ‘proximate causation’, etc.
The fact is that the new connective is not reflexive, that is, A ∋ A holds.
According to our classification, this system counts as a non-reflexive logic. New-
ton da Costa has other accounts on non-reflexive logics and identity, which can
be seen in [1], many of them in superposition with that was presented above.
6
Is identity really fundamental?
At least as present in every logical system, I guess no. The formation of our
theories and conceptions (so as our metaphysics) vary in time, from culture to
culture, and from different ways of conceptualization the surrounding world. I
mean ‘to conceptualize’ rather than ‘to perceive’, for a reasonable theory should
be the product of reasoning, and not only of just feelings. And, fundamentally,
we should take into account the difficulties in putting our claims in agreement
with those of the scientific community, mainly when it departs from the stan-
dards. Anyway, we need also to be aware of Martin Gardner’s warning of not
entering in the rol of the pseudo-scientists who think that they (and only they)
are right and all the others are wrong (see the first chapter of his book [14]).
So, in questioning a notion that is impregnated in our logic, our mathematics,
and our standard physics, for not speaking of our standard metaphysics, such
as identity, we need to be quite sure that the venture is a worthwhile endeavor.
And we are quite sure that this is the case with STI, mainly in considering
13
quantum mechanics.11
But here we need to be succinct. So, let us just make reference to an in-
teresting paper advanced by Otavio Bueno [2] where he try to safeguard the
fundamentality of the notion of identity. Unnecessary to say that he is thinking
of STI. An answer to Bueno was given in [24], so it not necessary to repeat the
discussion here. Thus, let me take this opportunity to advance some particular
further reflections on the reluctance some people has in questioning standard
identity.
One of the arguments is that we need identity to elaborate any theory, in
particular a theory where identity doesn’t hold for some entities. So, identity
could not be dispensed with. This rests in a clear confusion between levels of
languages. For instance, think of paraconsistent logics. These are logics where
the explosion rule doesn’t work unreservedly; there may be situations where
A and ¬A do not conduce to a trivialization (every well formed formula is a
theorem). In particular, by considering the paraconsistent negation (there are
several), the principle of non-contradiction ¬(A ∧ ¬A) is not universally valid.
Anyway, in building these systems,12 we do assume that the principle holds in
the metalanguage (or, if we would be more precise, in the metalogic). Really, no
one would suggest that some expression is a formula and simultaneously that it
is not, or that a certain sentence is a theorem and that it is not.
The same could be said about mathematics. Don’t we need to know what
is two in order to try to find a definition of ‘two’ ? Here, the differences among
levels of languages show their importance. We start reasoning in a rather in-
formal way, almost in a constructive way, where informal ideas are articulated,
combined and developed. With the progress of reasoning, we may abstract some
ideas and, in a limit, to arrive at strong systems (such as the ZFC set theory)
where we can, so to say, ‘reconstruct’ the notions we have used informally, such
as of the number two, so as the basic logical rules. This was called the Principle
of Constructivity by N. da Costa (see [4, p.57] and [23, p.43] for a discussion).
That is, we start with a free capacity of creative heuristics, but step by step we
will conforming our stuff to our ways of conceptualization. Thus, there is no
reason to say that we really can elaborate systems where the standard notion of
identity is questioned, as the principle of non-contradiction is in some systems.
7
Philosophy of non-reflexive logics
There are other interesting philosophical questions related to non-reflexive log-
ics. Let us consider those logics (like Schrödinger logics) that restrict the ap-
plicability of the notion of identity, so accepting that identity doesn’t apply to
some objects of the domain of the discourse. Da Costa and Bueno [8] point to
11There are also in some Eastern philosophies strong motivations for looking for alternative
notions than standard identity. It is remarkable that Schrödinger and H. Weyl based some of
their claims about identity in such views (again, see [12]).
12As said before there are plenty of paraconsistent systems, all of them with one sole finality:
to break the explosion rule.
14
the some questions that apparently provide puzzles to this kind of logic. In this
section, we provide a way to answer them into the scope of the non-reflexive
philosophy.
The first question concerns quantization. As they remark, in ‘reflexive log-
ics’, such as classical logic, there is an equivalence between ‘all F s are Gs’ and
‘each F is a G’, both symbolized by ∀x(F (x) → G(x)). Then, they note that
in order to quantify over each object of the domain, these objects need to be
identified, so identity needs to apply to them. Due to the identification be-
tween ‘each’ and ‘all’, according to these authors, we would be in trouble in
trying to quantify over objects that don’t have identity, for the standard def-
inition of satisfaction (so as denotation and truth) does require that identity
holds unreservedly. So, they say, a non-reflexive semantics cannot be given to
a non-reflexive logic, since for a faithful semantics, identity should not be avail-
able in the metalanguage where the semantics is to be developed. They call this
question the vicious circle of non-reflexivity.
Secondly, they say that non-reflexive logics distinguish between objects that
do have identity from those that do not, so “[i]t seems reasonable to take any
object of the first category from those of the second [. . . ] this is precisely the
sort of thing that cannot be said in a non-reflexive logic”. They call this the
paradox on non-reflexivity.
Let us address to these questions, starting with the last one. Within the
scope of non-reflexive ideas, we see that identity is not strictly necessary to
strongly distinguish among classes of objects. Suppose we have a molecule of
sulfuric acid, H2SO4. Notice that we have four Oxygen atoms, but cannot
distinguish them. We also cannot say that they are different under STI, for this
would require a property which applies to one of them but not to the others,
and of course there are none. So, it is enough to work with a weaker notion
of discernibility
(and indiscernibility), which is not necessarily equivalent to
identity. The four Oxygen atoms are indiscernible, and that fits better what
physics assume. As we have seen, this is achieved in the theory of quasi-sets
and in the higher-order Schrödinger logics.
In what concerns their exemple, that of discerning between entities that have
identity from those that do not, let us remark that we can discern between a
group of electrons and a group of chairs (which, by hypothesis, obey the standard
rules of identity), yet we cannot discern the electrons from one each other. No
problem here. In a non-reflexive setting, such as in the theory of quasi-sets, elec-
trons, protons and other quantum objects can be classified among the m-objects,
while chairs can be represented by M -objects. Quasi-set theory can indeed be
used as the metamathematics for a non-reflexive semantics for non-reflexive log-
ics, as we have seen before. So, with an adequate metamathematics, there is
neither paradox of non-reflexivity nor the vicious circle of non-reflexivity; suffice
to consider a non-reflexive mathematics acting in the metalanguage. Notice that
these authors consider just classical logic and standard semantics, although we
clearly are in a situation where these frameworks are being questioned.
Of course da Costa and Bueno could argue agains my argument by saying
that in proposing a quasi-set semantics for a non-reflexive logic, I am begging
15
the question, since I need to ‘create’ the theory of quasi-sets first and this cannot
be done without identity. This, again, is not fair. As we have seen above, using
their own Principle of Constructivity we can build such a system first assuming
identity in the ‘constructive stage’ and later bypassing it, as paraconsistent
logics did with the principle of non-contradiction (for more details on this, see
[23, Chap. 3]).
But the most interesting question is of course the problem of quantification.
Again, the authors are grounded on classical logic, assuming the equivalence
between ‘each’ and ‘all’ in the sense seem above. This is one way to understand
quantification. In a non-reflexive setting, of course, we are not obligated to
assume this ‘classical’ equivalence. It makes sense to speak of ‘all’ objects (even
m-objects) of a kind, but this doesn’t imply that we need to identity each of
them one by one. This can be seen, again, in the quantum contexts. Suppose a
Helium atom in its fundamental state (less energy). It has two electrons in an
entangled state, which means that there is no way to distinguish between them
or to take the states as separated. Important to remark that this is not a purely
epistemological impediment, but as far as quantum mechanic is believable, it is
an ontological problem; we really are dealing with entities that depart from the
standard objects of our surroundings. So, in the quantum realm there is no the
alleged equivalence between ‘all’ and ‘each’. Non-reflexive semantics gives sense
to that.
Notice that although we cannot discern electrons by pointing which is which,
we surely can say that all of them (of a certain class) have such and such
characteristics, like to share an entangled state. It is not necessary (by the way,
it is impossible) to speak of ‘this’ electron in distinction to ‘that’ electron. The
most we can say, let me emphasize, is that one of them has, say spin UP in a
given direction while the other one has spin DOWN in the same direction, but
never to determine which is which; by the way, this question doesn’t have any
sense at all.
So, how could we say that ‘some’ electrons (or protons, or neutrons, or an
H atom, or a water molecule, whatever quantum system you wish to consider)
are so and so? Without identity, we can ground our language on the notion of
cardinal (better, quasi-cardinals), as in quasi-set theory. Remember that the
notion of quasi-cardinal is primitive in Q, although it can be defined for finite
qsets. For instance, take again the He atom as above and the property P ‘x
is an electron (of He) and has spin UP in the z-direction’. As said before,
we know that just one of them has this property, but (according to quantum
mechanics) we cannot say which one. But, in Q, let e stand for ‘electron’. We
can form the strong singleton of e, termed JeKHe meaning one electron of the
He atom,13 but we cannot give him (significantly) a proper name, for we need
identity to make sense of that. This qset has q-cardinal 1 (as it follows from Q)
and its intension act as a definite description, but it cannot be eliminated by a
contextual definition (in Russell’s sense) due to the lack of identity. Really, there
13Of course we have the same problem in identifying the He atom, which is also a quantum
object, but we leave this point out.
16
will be just two of such strong singletons, and they are perfectly indiscernible;
they count as two, but we cannot point to any difference.
The same can be done with more than one m-object, say n of them. So,
having a qset with q-cardinal m > n, using a similar device in terms of q-
cardinals, we really can speak of n of them satisfying a certain characteristics,
in the same vein that we say that in the 2p shell of a Na atom, there are six
electrons, even without being able to say which of the 11 electrons of the atom
are those of the level 2p.14
Notice that the sentence ‘those electrons among
the 11 that belong to the 2p shell’ doesn’t identity a sub-collection, for we
really cannot identify these six electrons from the remaining ones, despite their
peculiar characteristics.
This is a key point we need to insist. Suppose you have a box with 11
indiscernible balls and that you are in a completely dark room. You need to
separate 6 of them. You can do it, but you don’t know which ones you are
taking. Only later, with light or other devices (you can make a physical mark
on the six) you can identify them. But this cannot be done with electrons or
other quantum systems! So, you can say that the Na atom has six electrons in
the 2p shell, but there are no means to say which ones are there. I strongly
hope that this shows you that you are not discerning them by a property, since
you don’t know which ones are your six.
So, these criticisms don’t stick to non-reflexive logics. Furthermore, as we
see, non-reflexive logics are really a class of well characterized logical systems,
with adequate syntactical and semantical counterparts, so they can be legiti-
mately classified in the class of heterodox logics.
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